Convexity of the free boundary for two-dimensional Réthy flows

Yukun Shangguan; Yukun Shangguan

doi:10.3934/math.20261396

AIMS Mathematics

2026, Volume 11, Issue 11: 36065-36079. doi: 10.3934/math.20261396

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Research article

Convexity of the free boundary for two-dimensional Réthy flows

Yukun Shangguan ^,

School of Mathematics, North University of China, Taiyuan 030051, Shanxi, China

Received: 28 July 2026 Revised: 11 October 2026 Accepted: 22 October 2026 Published: 03 November 2026
MSC : 35Q31, 35J25, 76B10, 35Q35

This paper investigates the convexity of the free boundaries for two-dimensional symmetric Réthy flows. Building on the known existence theory, we consider a steady, incompressible, irrotational flow past an obstacle. Under the assumptions that the upper nozzle wall is horizontal and that the obstacle profile satisfies suitable convexity conditions, we establish a relation between streamline curvature and the normal derivative of the flow speed. Using the subharmonicity of the squared speed and the maximum principle, we prove that the speed in the fluid interior is strictly less than its constant value on the free boundaries. Finally, Hopf's boundary-point lemma and the curvature–speed relation imply that the upper free-boundary graph is strictly convex and that the lower free-boundary graph is strictly concave.
- Réthy flow,
- free boundary,
- convexity,
- maximum principle,
- Hopf lemma
Citation: Yukun Shangguan. Convexity of the free boundary for two-dimensional Réthy flows[J]. AIMS Mathematics, 2026, 11(11): 36065-36079. doi: 10.3934/math.20261396

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Abstract

This paper investigates the convexity of the free boundaries for two-dimensional symmetric Réthy flows. Building on the known existence theory, we consider a steady, incompressible, irrotational flow past an obstacle. Under the assumptions that the upper nozzle wall is horizontal and that the obstacle profile satisfies suitable convexity conditions, we establish a relation between streamline curvature and the normal derivative of the flow speed. Using the subharmonicity of the squared speed and the maximum principle, we prove that the speed in the fluid interior is strictly less than its constant value on the free boundaries. Finally, Hopf's boundary-point lemma and the curvature–speed relation imply that the upper free-boundary graph is strictly convex and that the lower free-boundary graph is strictly concave.

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